Vol. 26, No. 3, 1968

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 294: 1
Vol. 293: 1  2
Vol. 292: 1  2
Vol. 291: 1  2
Vol. 290: 1  2
Vol. 289: 1  2
Vol. 288: 1  2
Vol. 287: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Subscriptions
Editorial Board
Officers
Special Issues
Submission Guidelines
Submission Form
Contacts
Author Index
To Appear
 
ISSN: 0030-8730
Bases in Hilbert space

William Jay Davis

Vol. 26 (1968), No. 3, 441–445
Abstract

A sequence (xi) of elements of a Hilbert space, , is a basis for if every h ∈ℋ has a unique, norm-convergent expansion of the form h = aixi, where (ai) is a sequence of scalars. The sequence is minimal if there exists a sequence (yi) ⊂ℋ such that (xi,yj) = δij. Every basis is minimal, and the sequence (ai) in the expansion of h (above) is given by ai = (h,yi). In this paper, we restrict our attention to real Hilbert space.

We derive, from classical characterizations of bases in B-spaces, criterea for (xi) to be a basis for , as well as for (xi) to be minimal in . We show that the sequence is minimal if and only if there are sequences (gi) ⊂ℋ whose Gram matrices have a prescribed form. Similar conditions are obtained for (xi) to be a basis for .

Mathematical Subject Classification
Primary: 46.15
Milestones
Received: 13 December 1966
Published: 1 September 1968
Authors
William Jay Davis